Types of and Purposes for Elementary Mathematical Writing

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Disclaimer

This material is based on work supported by the National Science Foundation under Grant No. 1545908, Task Force on Conceptualizing Elementary Mathematical Writing: Implications for Mathematics Education Stakeholders. The task force met at the University of Connecticut in October 2015, and this document was released April 2016. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

Citation Information

This report is in the public domain. Although permission to reprint this publication is not necessary, the citation should be: Casa , T. M., Firmender, J. M., Cahill, J., Cardetti, F., Choppin, J. M., Cohen, J., ... Zawodniak, R. (2016). Types of and purposes for elementary mathematical writing: Task force recommendations. Retrieved from .

Task Force Members

Tutita M. Casa (Director)

University of Connecticut

June Cahill

Hartford CT Public Schools

Jeffrey M. Choppin

University of Rochester

Shelbi Cole

Student Achievement Partners

Juanita Copley

University of Houston (Emerita)

Jack Dieckmann

Stanford University

M. Katherine Gavin

University of Connecticut

Karen S. Karp

Johns Hopkins University

Judit N. Moschkovich

University of California at Santa Cruz

Natalie G. Olinghouse

University of Connecticut

Erin Price

Fairfax County VA Public Schools

Betsy Rupp Fulwiler

Science Education Consultant

Ryley Zawodniak

Mansfield CT Public Schools

Janine M. Firmender (Co-Director)

Saint Joseph's University

Fabiana Cardetti

University of Connecticut

Jeremy Cohen

Union NJ Public Schools, Fordham University

Madelyn W. Colonnese

University of Connecticut

Michael DiCicco

Mansfield CT Public Schools

Jennifer Dorl

West Hartford CT Public Schools

Michael A. Hebert

University of Nebraska-Lincoln

Erika LaBella

Mansfield CT Public Schools

Karen Moylan

Mansfield CT Public Schools

Sarah R. Powell

University of Texas at Austin

David K. Pugalee

University of North Carolina at Charlotte

Linda J. Sheffield

Northern Kentucky University (Emerita)

"At its best, writing is learning."1 But what types of writing have the potential to further elementary students' learning of mathematics?

The Elementary Mathematical Writing Task Force outlines in this document recommendations for four types of mathematical writing and the multiple purposes of each type. Importantly, the task force distinguishes mathematical writing from writing about mathematics. That is, while mathematics content knowledge was at the forefront of the task force's discussions, writing was seen as a means through which students can develop and express their mathematical reasoning.

Mathematical Writing: Important, But Undefined

Teachers long have been encouraged to engage students in communication as a crucial component of learning mathematics. Writing in mathematics is considered beneficial for students because, through writing, "they communicate to learn mathematics, and they learn to communicate mathematically."2 Having students write about their ideas helps them learn content3,4,5 and develops their problem-solving abilities.6 The process of writing also allows students to reason immediately and visually about the correctness of their solution.7

The National Council of Teachers of Mathematics has established that "writing is another important component of the discourse."8 However, until now, descriptions of the types of and purposes for mathematical writing have fallen short in adequately informing instructional practices. For example, national documents suggest that students use "written communication"9 and "express themselves increasingly clearly and coherently."10 The National Research Council states that students should "justify and explain ideas in order to make their reasoning clear,"11 while the Common Core State Standards call for students to "construct viable arguments and critique the reasoning of others."12

Given this rather limited clarity as to the purposes for mathematical writing, it is not surprising that writing has been implemented in a variety of ways in mathematics instruction.13 Students have been asked, for example, to develop creative pieces,14,15 engage in expository writing,16,17 record one's feelings about mathematics,18 compose a biography of a famous mathematician,19 reflect on an activity,20,21,22 and take notes.23 Although these types of writing may benefit students' literacy, what types of writing are recommended for elementary

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students, starting in kindergarten,24 to lay the foundation for future reasoning and proof writing? Providing a framework for the types of and purposes for mathematical writing enables elementary teachers to leverage writing for students' learning of mathematics, coaches to support teachers with its implementation, assessment developers to design and score such items, and curriculum developers to provide more opportunities for students to engage in this practice.

Clarifying Elementary Mathematical Writing

The Elementary Mathematical Writing Task Force convened in Fall 2015 to address the lack of clarity about the types of and purposes for mathematical writing. This meeting brought together a diverse group of school- and universitybased experts to ensure a comprehensive perspective and offer recommendations that would attend to the needs of various stakeholder groups and students.

Task force members collectively represented the fields of mathematics education, mathematics, and writing education; had expertise across the elementary grades (K-6); were knowledgeable about particular student groups, including English language learners (ELLs), students who have learning difficulties, and/or those students identified as gifted; and had experience teaching writing and authoring assessments and curricula that include mathematical writing.

The primary goals for the Elementary Mathematical Writing Task Force were to: Consider various purposes for which students might be asked to write in their mathematics class; Reach a consensus about the types of elementary mathematical writing that are reflective of these multiple purposes and recommend the types that leverage students' mathematical learning; and Account for perspectives from multiple stakeholders, evidence of students' potential for writing productively in mathematics, and multiple sets of curriculum standards.

Task force members first identified a mathematical writing artifact that represented their original views on mathematical writing and shared it with the task force. Next, they worked collaboratively in small groups -- initially with fellow members who had similar professional backgrounds and then with

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members specializing in other areas of expertise -- to analyze writing-related items. These items included national standards from mathematics education and writing education, as well as samples of students' mathematical writing. Each small group developed working descriptions for the types of and purposes for mathematical writing and then refined them throughout the meeting. Each iteration of the descriptions of the types of and purposes for elementary mathematical writing were presented to the entire task force to facilitate discussions that ultimately led to the recommendations described herein.

Overarching Goals for Elementary Mathematical Writing

Two overarching goals for elementary mathematical writing were central to the task force's recommendations: for students to reason mathematically and to communicate ideas. Reasoning and communication have long been emphasized in the teaching and learning of mathematics.25,26,27,28,29 Together, these overarching goals have the potential to provide students with an authentic way to engage with mathematics.

Students who reason mathematically "tend to note patterns, structure, or regularities in both real-world situations and symbolic objects; they ask if those patterns are accidental or if they occur for a reason; and they conjecture and prove."30 All recommended types of and purposes for elementary mathematical writing should ultimately serve to engage students in reasoning. (Note: Some purposes for mathematical writing that minimize mathematical reasoning, such as composing a mathematical biography or a narrative describing one's feelings about math, are not addressed in this document.)

Students who communicate mathematically in writing can go beyond using words, phrases, and sentences because "mathematics is so often conveyed in symbols."31 Like symbols, other representations, such as drawings, tables, and graphs, can serve to communicate mathematical ideas.32 It is important that any symbols and representations used in mathematical writing should support students' mathematical ideas.

Types of and Purposes for Elementary Mathematical Writing

The Elementary Mathematical Writing Task Force recommends four types of mathematical writing. These types of writing serve different purposes within the

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overarching goals of having students reason and communicate mathematically (Figure 1). Each type of writing (highlighted in dark red) defines an overall category that comprises similar purposes. The purpose(s) (bulleted in black text) provide specific reason(s) for engaging students in mathematical writing.

Figure 1. Overview of the overarching goals and the types of and purposes for elementary mathematical writing

The types of and purposes for elementary mathematical writing are further described in the remainder of this report. Several considerations guided the task force's recommendations. First and foremost, all elementary students should write mathematically and "reasonable and appropriate accommodations [should] be made as needed to promote access and attainment for all students,"33 including those with learning difficulties and those who are English language learners. Additional considerations include the following:

Writing develops across multiple continua. First, it is reasonable to assume that younger students' writing would be less sophisticated than that of older students. For example, kindergarteners may use invented spelling, or

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teachers may transcribe ideas conveyed in students' notations and representations for others to read. Second, it would be expected that students' writing would develop over the course of any given year through multiple experiences with a given writing type and purpose. Third, some writing may develop using writing processes (plans, drafts, revisions, final product). However, with mathematical writing, the goal for writing may not require the full trajectory of the complete writing process, and the end result may not be a final published piece.

The audience influences students' mathematical writing. Often, the teacher serves as the students' audience, albeit not explicitly. Students can write to an outside or authentic audience, such as their peers, family members, and community members. In some cases, students might serve as their own audience. Particular expectations may be present when writing to an outside audience, as on assessments.

Mathematical writing may take multiple forms. The types of and purposes for writing can be captured in a variety of forms, such as students' notations, a paragraph, or a letter (see the appendix). However, these forms may be more formal and imply that students conform to literacy conventions. The forms that teachers or students decide to utilize, if any, should not override the overarching goals or the types of and purposes for elementary mathematical writing.

Notes About the Student Writing Samples

The student writing samples we include here generally illustrate the types of and purposes for mathematical writing. The samples are not meant to be exemplars.

The samples were gathered across various projects, districts, and grades resulting from task force members' work. Most samples were created prior to the identification of the task force recommendations.

The samples represent writing from across the school year. The amount of instructional guidance and support within the writing prompt for the development of student writing also varied.

The set was selected, in part, to signify characteristics of mathematical writing, such as the use of symbols, representations, and vocabulary.

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